In Project 1.4, we saw that Po invested in a bank account earning a simple amual interest rate of r compounded n times per year for t years is worth P(t) = R(1+E). In many precalculus and calculus books, it is shown that, as n x, this equation becomes P(t) = Pe, which is the solution of a discrete exponential model (not base e) can be expressed e) and vice versa, which matches exactly at times corresponding to the whole numbers of time steps. Thus, when convenient to do so, a discrete exponential situation can be modeled with differential equations which give accurate answers where the domains make =rP. We discussed in the text that as a continuous model (base sense. 1. If a bank account has a simple annual rate of r percent compounded monthly, find the appropriate interest rate to use for a continuous model 2. People with sizable investments often use interest income to live on either partially wholly. Suppose that an account earns at an annual rate ofr percent compounded tinuously and a person is drawing income ofH dollars per year withdrawn continuously (impossible, but a modeling assumption). Use phase line analysis to analyze the behavior of the account. Discuss the meaning of any equilibrium points and their stability. If T = 10% ( a reasonable rate for long-term stock investments). and H $10.000, how or con- long should an initial investment of $50.000 be left untouched so that when withdrawals is not depleted? begin the capital 3. In general. suppose that F t represents the net flow of money into and out of an account over time, not counting the income due to interest. Verify that the goveming differential equation has the form rP+ F( t). Express the solution of this differential equation in termms of an integral. (Hint: This is a linear differential equation with a forcing term and can be treated as a linear diferential equation with nonconstant coeffficients.) In Project 1.4, we saw that Po invested in a bank account earning a simple amual interest rate of r compounded n times per year for t years is worth P(t) = R(1+E). In many precalculus and calculus books, it is shown that, as n x, this equation becomes P(t) = Pe, which is the solution of a discrete exponential model (not base e) can be expressed e) and vice versa, which matches exactly at times corresponding to the whole numbers of time steps. Thus, when convenient to do so, a discrete exponential situation can be modeled with differential equations which give accurate answers where the domains make =rP. We discussed in the text that as a continuous model (base sense. 1. If a bank account has a simple annual rate of r percent compounded monthly, find the appropriate interest rate to use for a continuous model 2. People with sizable investments often use interest income to live on either partially wholly. Suppose that an account earns at an annual rate ofr percent compounded tinuously and a person is drawing income ofH dollars per year withdrawn continuously (impossible, but a modeling assumption). Use phase line analysis to analyze the behavior of the account. Discuss the meaning of any equilibrium points and their stability. If T = 10% ( a reasonable rate for long-term stock investments). and H $10.000, how or con- long should an initial investment of $50.000 be left untouched so that when withdrawals is not depleted? begin the capital 3. In general. suppose that F t represents the net flow of money into and out of an account over time, not counting the income due to interest. Verify that the goveming differential equation has the form rP+ F( t). Express the solution of this differential equation in termms of an integral. (Hint: This is a linear differential equation with a forcing term and can be treated as a linear diferential equation with nonconstant coeffficients.)
